Mathematics for Competitive Exam Success

Mathematics

3D Geometry Applications of Derivatives Binomial Theorem Circles Complex Numbers Conic Sections Coordinate Geometry Differential Equations Differentiation Integration Limits & Continuity Matrices & Determinants Permutations & Combinations Probability Quadratic Equations Sequences & Series Sets & Relations Straight Lines Trigonometry Vector Algebra

Q. Find the minimum value of f(x) = x^2 - 4x + 7. (2021)

A. 3
B. 5
C. 4
D. 6

Solution

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.

Correct Answer: A — 3

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Q. Find the minimum value of f(x) = x^2 - 4x + 7. (2021) 2021

A. 3
B. 5
C. 4
D. 6

Solution

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.

Correct Answer: A — 3

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Q. Find the minimum value of the function f(x) = 2x^2 - 8x + 10. (2022)

A. 2
B. 4
C. 6
D. 8

Solution

The minimum occurs at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 6.

Correct Answer: B — 4

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Q. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.

A. y = 2e^(4x)
B. y = e^(4x)
C. y = 4e^(x)
D. y = 2e^(x)

Solution

The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).

Correct Answer: A — y = 2e^(4x)

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Q. Find the particular solution of dy/dx = 4y, given y(0) = 2.

A. y = 2e^(4x)
B. y = e^(4x)
C. y = 4e^(2x)
D. y = 2e^(x/4)

Solution

The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).

Correct Answer: A — y = 2e^(4x)

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Q. Find the point of inflection for f(x) = x^3 - 6x^2 + 9x. (2022)

A. (1, 4)
B. (2, 3)
C. (3, 0)
D. (0, 0)

Solution

f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. f(2) = 3.

Correct Answer: C — (3, 0)

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Q. Find the real part of the complex number 4 + 5i. (2023)

A. 4
B. 5
C. 9
D. 0

Solution

The real part of the complex number 4 + 5i is 4.

Correct Answer: A — 4

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Q. Find the roots of the equation x² + 2x - 8 = 0. (2022)

A. -4 and 2
B. 4 and -2
C. 2 and -4
D. 0 and 8

Solution

Factoring gives (x + 4)(x - 2) = 0, hence the roots are 4 and -2.

Correct Answer: B — 4 and -2

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Q. Find the slope of the tangent line to f(x) = 2x^3 - 3x^2 + 4 at x = 1. (2021)

A. 1
B. 2
C. 3
D. 4

Solution

f'(x) = 6x^2 - 6. f'(1) = 6(1)^2 - 6 = 0.

Correct Answer: B — 2

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Q. Find the solution of the differential equation dy/dx = y^2.

A. y = 1/(C - x)
B. y = C/(x - 1)
C. y = Cx
D. y = e^(x)

Solution

This is a separable equation. Integrating gives y = 1/(C - x).

Correct Answer: A — y = 1/(C - x)

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Q. Find the solution of the differential equation y' = 3y + 6.

A. y = Ce^(3x) - 2
B. y = Ce^(3x) + 2
C. y = 2e^(3x)
D. y = 3Ce^(x)

Solution

This is a linear first-order equation. The integrating factor is e^(3x). The solution is y = Ce^(3x) + 2.

Correct Answer: B — y = Ce^(3x) + 2

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Q. Find the solution of the equation dy/dx = y^2 - 1.

A. y = tan(x + C)
B. y = C/(1 - Cx)
C. y = 1/(C - x)
D. y = C/(x + 1)

Solution

This is a separable equation. The solution is y = tan(x + C).

Correct Answer: A — y = tan(x + C)

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Q. Find the term containing x^3 in the expansion of (x - 1)^5.

A. -5
B. 10
C. -10
D. 5

Solution

The term containing x^3 is C(5,3) * x^3 * (-1)^2 = 10 * x^3 * 1 = 10.

Correct Answer: C — -10

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Q. Find the term independent of x in the expansion of (x^2 - 2x + 3)^4. (2022)

A. 81
B. 108
C. 54
D. 27

Solution

The term independent of x occurs when the powers of x cancel out. The term is 81.

Correct Answer: A — 81

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Q. Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)

A. -15
B. 10
C. 5
D. 0

Solution

The term independent of x occurs when the powers of x cancel out. The term is C(5,2)(-3)^2(1)^3 = 45.

Correct Answer: A — -15

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Q. Find the value of (3 + 2)^3 using the binomial theorem.

A. 25
B. 27
C. 30
D. 32

Solution

Using the binomial theorem, (3 + 2)^3 = C(3,0) * 3^3 * 2^0 + C(3,1) * 3^2 * 2^1 + C(3,2) * 3^1 * 2^2 + C(3,3) * 3^0 * 2^3 = 27 + 54 + 36 + 8 = 125.

Correct Answer: B — 27

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Q. Find the value of k for which the equation x² + 4x + k = 0 has no real roots. (2020)

A. -5
B. -6
C. -4
D. -3

Solution

The discriminant must be negative: 4² - 4*1*k < 0, which gives k > 4, so the minimum value is -6.

Correct Answer: B — -6

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Q. Find the value of k for which the equation x² + kx + 16 = 0 has equal roots. (2022)

A. -8
B. -4
C. 4
D. 8

Solution

For equal roots, the discriminant must be zero: k² - 4*1*16 = 0, thus k² = 64, k = ±8. The value of k can be -8.

Correct Answer: A — -8

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Q. Find the value of k if the equation x² + kx + 16 = 0 has no real roots. (2022)

A. k < 8
B. k > 8
C. k < 0
D. k > 0

Solution

For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, which gives k > 8.

Correct Answer: B — k > 8

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Q. Find the y-intercept of the line 4x + y - 8 = 0.

A. 8
B. 4
C. 2
D. 0

Solution

Setting x = 0 in the equation gives y = 8. Therefore, the y-intercept is 8.

Correct Answer: A — 8

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Q. Find ∫ (5x^4) dx. (2020)

A. x^5 + C
B. x^5 + 5C
C. x^5 + 1
D. 5x^5 + C

Solution

The integral is (5/5)x^5 + C = x^5 + C.

Correct Answer: A — x^5 + C

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Q. Find ∫ (6x^2 - 4) dx. (2019)

A. 2x^3 - 4x + C
B. 2x^3 - 2x + C
C. 2x^3 - 4 + C
D. 3x^3 - 4x + C

Solution

The integral is (6/3)x^3 - 4x + C = 2x^3 - 4x + C.

Correct Answer: A — 2x^3 - 4x + C

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Q. Find ∫ e^x dx. (2022)

A. e^x + C
B. e^x
C. x e^x + C
D. ln(e^x) + C

Solution

The integral of e^x is e^x + C.

Correct Answer: A — e^x + C

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Q. For the equation x² + 6x + k = 0 to have real roots, what is the minimum value of k? (2021)

A. -9
B. -6
C. 0
D. 6

Solution

The discriminant must be non-negative: 6² - 4*1*k ≥ 0, which gives k ≤ 9, so the minimum value is -9.

Correct Answer: A — -9

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Q. For the function f(x) = -x^2 + 4x + 1, find the x-coordinate of the vertex. (2023)

A. 2
B. 4
C. 1
D. 3

Solution

The vertex x-coordinate is given by -b/(2a) = -4/(2*-1) = 2.

Correct Answer: A — 2

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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the critical points. (2022)

A. (0, 0)
B. (1, 5)
C. (2, 0)
D. (3, 3)

Solution

Set f'(x) = 0. f'(x) = 6x^2 - 18x + 12 = 0. Critical points are x = 2.

Correct Answer: C — (2, 0)

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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the local maxima. (2023) 2023

A. (1, 5)
B. (2, 6)
C. (3, 3)
D. (0, 0)

Solution

Find f'(x) = 6x^2 - 18x + 12, set to 0. Local maxima at x = 2 gives f(2) = 6.

Correct Answer: B — (2, 6)

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Q. For the function f(x) = 3x^2 - 12x + 7, find the minimum value. (2022)

A. -5
B. -4
C. -3
D. -2

Solution

The vertex occurs at x = 2. f(2) = 3(2^2) - 12(2) + 7 = -5.

Correct Answer: B — -4

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Q. For the function f(x) = 3x^2 - 12x + 7, find the x-coordinate of the vertex. (2022)

A. 1
B. 2
C. 3
D. 4

Solution

The x-coordinate of the vertex is given by x = -b/(2a) = 12/(2*3) = 2.

Correct Answer: B — 2

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Q. For the function f(x) = 3x^2 - 12x + 9, find the vertex. (2021)

A. (2, 3)
B. (3, 0)
C. (0, 9)
D. (1, 6)

Solution

The vertex occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.

Correct Answer: A — (2, 3)

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Mathematics

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